Past

In this talk I will discuss "motivic" Donaldson-Thomas invariants, following the now not-so-recent paper of Kontsevich and Soibelman on this subject. I will, in particular, present some understanding of the mysterious notion of "orientation data," and present some recent work. I will of course do my best to make this talk "accessible," though if you don't know what a scheme or a category is it will probably make you cry.

The second talk will present conjectural motivic generalizations

of ADHM sheaf invariants as well as their wallcrossing formulas.

It will be shown that these conjectures yield recursive formulas

for Poincare and Hodge polynomials of moduli spaces of Hitchin

pairs. It will be checked in many concrete examples that this recursion relation is in agreement with previous results of Hitchin, Gothen, Hausel and Rodriguez-Villegas.

The first talk will present a construction of equivariant

virtual counting invariants for certain quiver sheaves on a curve, called ADHM sheaves. It will be shown that these invariants are related to the stable pair theory of Pandharipande and Thomas in a specific stability chamber. Wallcrossing formulas will be derived using the theory of generalized Donaldson-Thomas invariants of Joyce and Song.

This talk will begin with an introduction to calibrations and calibrated submanifolds. Calibrated geometry generalizes Wirtinger's inequality in Kahler geometry by considering k-forms which are analogous to the Kahler form. A famous one-line proof shows that calibrated submanifolds are volume minimizing in their homology class. Our examples of manifolds with a calibration will come from complex geometry and from manifolds with special holonomy.

We will then discuss the deformation theory of the calibrated submanifolds in each of our examples and see how they differ from the theory of complex submanifolds of Kahler manifolds.

The presence and flow of fluid inside a crack within a solid causes deformation of the solid which in turn influences the flow of the fluid.

This coupled fluid-solid problem will be discussed in the context of dyke propagation and hydrofracture. The background material will be discussed in detail and some applications to specific geometries presented.

"We investigate a construction of a Skorokhod embedding due to Root (1969), which has been the subject of recent interest for applications in Mathematical Finance (Dupire, Carr & Lee), where the construction has applications for model-free pricing and hedging of variance derivatives. In this context, there are two related questions: firstly of the construction of the stopping time, which is related to a free boundary problem, and in this direction, we expand on work of Dupire and Carr & Lee; secondly of the optimality of the construction, which is originally due to Rost (1976). In the financial context, optimality is connected to the construction of hedging strategies, and by giving a novel proof of the optimality of the Root construction, we are able to identify model-free hedging strategies for variance derivatives. Finally, we will present some evidence on the numerical performance of such hedges. (Joint work with Jiajie Wang)"

This talk will be based on the article arXiv:1006.2788.

It is well-known that Morrey's quasiconvexity is closely related to gradient Young measures,

i.e., Young measures generated by sequences of gradients in

$L^p(\Omega;\mathbb{R}^{m\times n})$. Concentration effects,

however, cannot be treated by Young measures. One way how to describe both oscillation and

concentration effects in a fair generality are the so-called DiPerna-Majda measures.

DiPerna and Majda showed that having a sequence $\{y_k\}$ bounded in $L^p(\Omega;\mathbb{R}^{m\times n})$,$1\le p$ $0$.

Puck Rombach;

"Weighted Generalization of the Chromatic Number in Networks with Community Structure",

Christopher Lustri;

"Exponential Asymptotics for Time-Varying Flows,

Alex Shabala

"Mathematical Modelling of Oncolytic Virotherapy",

Martin Gould;

"Foreign Exchange Trading and The Limit Order Book"

The n-amalgamation property has recently been explored in connection with generalised imaginaries (groupoid imaginaries) by Hrushovski. This property is useful when studying models of a stable theory together with a generic automorphism, e.g.

elimination of imaginaries (e.i.) in ACFA may be seen as a consequence of 4-amalgamation (and e.i.) in ACF.

The talk is centered around 4-amalgamation of stably dominated types in algebraically closed valued fields. I will show that 4-amalgamation holds in equicharacteristic 0, even for systems with 1 vertex non stably dominated. This is proved using a reduction to the stable part, where 4-amalgamation holds by a result of Hrushovski. On the other hand, I will exhibit an NIP (even metastable) theory with 4-amalgamation for stable types but in which stably dominated types may not be 4-amalgamated.

The question in the title seems to be neglected in the studies of open dynamical systems. It occurred though that the features of dynamics may play a role comparable to the one played by the size of a hole. For instance, the escape through the smaller hole could be faster than through the larger one.

These studies revealed as well a new role of the periodic orbits in the dynamics which could be exactly quantified in some cases. Moreover, this new approach allows to characterize the elements of networks by their dynamical properties (rather than by static ones like centrality, betweenness, etc.)

Not only does the definition of an (abstract) saturated fusion system provide us with an interesting way to think about finite groups, it also permits the construction of exotic examples, i.e. objects that are non-realisable by any finite group. After recalling the relevant definitions of fusion systems and saturation, we construct an exotic fusion system at the prime 3 as the fusion system of the completion of a tree of finite groups. We then sketch a proof that it is indeed exotic by appealing to The Classification of Finite Simple Groups.

An important problem in algebra is the study of algebraic objects

defined over fields and how they behave under field extensions,

for example the Brauer group of a field, Galois cohomology groups

over fields, Milnor K-theory of a field, or the Witt ring of bilinear

forms over

a field. Of particular interest is the determination

of the kernel of the restriction map when passing to a field extension.

We will give an overview over some known results concerning the

kernel of the restriction map from the Witt ring of a field to the

Witt ring of an extension field. Over fields of characteristic

not two, general results are rather sparse. In characteristic two,

we have a much more complete picture. In this talk, I will

explain the full solution to this problem for extensions that are

given by function fields of hypersurfaces over fields of

characteristic two. An important tool is the study of the

behaviour of differential forms over fields of positive

characteristic under field extensions. The result for

Witt rings in characteristic two then follows by applying earlier

results by Kato, Aravire-Baeza, and Laghribi. This is joint

work with Andrew Dolphin.

Following work done by the 'Oxford Spies' we uncover more secrets of 'surface-active Agents'. In modern-day applications we refer to these agents as surfactants, which are now extensively used in industrial, chemical, biological and domestic applications. Our work focuses on the dynamic behaviour of surfactant and polymer-surfactant mixtures.

In this talk we propose a mathematical model that incorporates the effects of diffusion, advection and reactions to describe the dynamic behaviour of such systems and apply the model to the over-flowing-cylinder experiment (OFC). We solve the governing equations of the model numerically and, by exploiting large parameters in the model, obtain analytical asymptotic solutions for the concentrations of the bulk species in the system. Thus, these solutions uncover secrets of the 'surface-active Agents' and provide an important insight into the system behaviour, predicting the regimes under which we observe phase transitions of the species in the system. Finally, we suggest how our models can be extended to uncover the secrets of more complex systems in the field.